# IA-automorphism group of nilpotent group equals stability group of lower central series

## Contents

## Statement

### For an individual automorphism

Suppose is a nilpotent group. Then, the following are equivalent for an automorphism of :

- is an IA-automorphism of , i.e., it induces the identity map on the abelianization of .
- is a stability automorphism for the lower central series of .

Note that being nilpotent is important only in so far as it guarantees that the lower central series reaches the trivial subgroup. A slight variant of the statement would be true for non-nilpotent groups, but we wouldn't use the jargon of *stability automorphism*.

### For subgroups of the automorphism group

Suppose is a nilpotent group. The following subgroups of the automorphism group of are equal:

- The subgroup of IA-automorphisms of , i.e., automorphisms that induce the identity map on the abelianization of .
- The stability group (i.e., the group of stability automorphisms) for the lower central series of .

## Proof

### Background fact on iterated commutator mapping

The proof basically follows from the fact that for any positive integer , the left-normed iterated commutator gives a surjective -linear map from the abelianization of to the quotient group between successive members of the lower central series. Here, is the member of the lower central series of . Explicitly, it is a -linear map of abelian groups:

This mapping is canonical, hence covariant with automorphisms.

### (1) implies (2)

**Given**: An IA-automorphism of a nilpotent group .

** To prove**: induces the identity map on each of the successive quotients of the lower central series, i.e., for every , induces the identity map on each of the groups of the form where is the member of the lower central series of .

**Proof**: The covariance of the iterated commutator mapping with respect to , along with its surjectivity to , guarantees that since fixes pointwise, it also fixes the image pointwise.

### (2) implies (1)

**Given**: A nilpotent group . An automorphism induces the identity map on each of the successive quotients of the lower central series, i.e., for every , induces the identity map on each of the groups of the form where is the member of the lower central series of .

**To prove**: is an IA-automorphism of .

**Proof**: This follows directly from setting .